Seite - 225 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

A.1. DerivationofExtendedKalmanFilter Then finally the covariance matrix of the estimation error can be ex- pressedby [Pne(k)] =E [ [ene(k)][e n e(k)] T ] =E [(( [I]− [Kn(k)][Jn(k)])[enp(k)]− ς(k)[Kn(k)]) (( [I]− [Kn(k)][Jn(k)])[enp(k)]− ς(k)[Kn(k)])T] = ( [I]− [Kn(k)][Jn(k)])[Pnp(k)]([I]− [Kn(k)][Jn(k)])T +σ2 [Kn(k)][Kn(k)] T = [ Pnp(k) ]− [Kn(k)][Jn(k)][Pnp(k)] −[Pnp(k)][Jn(k)]T [Kn(k)]T +[Kn(k)][J] n (k) [ Pnp(k) ] [Jn(k)] T [Kn(k)] T +σ2 [Kn(k)][Kn(k)] T (A.14) The objective of EKF is to minimize the estimation error [ene(k)] re- garding the gain matrix [Kn(k)], which in other words, is equivalent to find an appropriate matrix [Kn(k)] minimizing the covariance ma- trix [Pne(k)]. Followingtheprincipleof [Kn(k)] = arg[K]min[P n e(k)]⇒ ∂ [Pne(k)] ∂ [Kn(k)] = 0, (A.15) thegainmatrix isobtainedas [Kn(k)] = [ Pnp(k) ] [Jn(k)] T ( [Jn(k)] [ Pnp(k) ] [Jn(k)] T +σ2 [IM] )−1 . (A.16) 225